Impact of widely used approximations to the G0 W0 method: An all-electron
perspective
Xin-Zheng Li,1 Ricardo Gómez-Abal,1 Hong Jiang,1, ∗ Claudia Ambrosch-Draxl,2 and Matthias Scheffler1
1
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195, Berlin, Germany
2
Chair of Atomistic Modelling and Design of Materials,
Montanuniversität Leoben, Franz-Josef-Straße 18, A-8700, Austria
(Received July 25, 2011)
Focussing on the fundamental band gaps in Si, diamond, BN, LiF, AlP, NaCl, CaSe, and GaAs,
and the semicore d-state binding energies in ZnS, ZnSe, ZnTe, CdS, CdSe, CdTe, and GaN, we
analyze the well-known discrepancies between the pseudopotential (PP) and all-electron (AE) G0 W0
results. Approximations underlying PP-G0 W0 , i.e., the frozen-core, the core-valence partitioning,
and the use of pseudo-wavefunctions, are separately addressed. The largest differences, of the order
of eV, appear in the exchange part of the self-energy and the exchange-correlation (xc) potential due
to the core-valence partitioning. These differences cancel each other and, in doing so, make the final
core-valence partitioning effect on the band gaps controllable when the semicore states are treated as
valence states. This cancellation, however, is incomplete for semicore d-state binding energies, due
to the strong interaction between these semicore states and the deeper core. The remaining error can
be reduced by treating the outermost two shells as valence shell. However, reliably describing these
many-body interactions at the G0 W0 level and providing benchmark results requires an all-electron
approach.
PACS numbers: 71.15.-m, 71.15.Ap, 71.10.-w
I.
INTRODUCTION
Many-body perturbation theory (MBPT) establishes
a formally exact framework for the interpretation of the
quasi-particle band structure in solids in terms of singleparticle excitations. Within this framework, the central
quantity is the Green function. Its poles in the complex frequency plane determine the excitation energies
of the system in terms of single electrons in formerly
unoccupied states or single holes in formerly occupied
states, as measured by inverse and direct photoemission,
respectively. Calculating the Green function, however,
requires the knowledge of another quantity, i.e., the selfenergy through the self-consistent solution of the Dyson
equation. Being a non-local, energy-dependent operator which contains the information about all many-body
interactions between electrons, the self-energy needs to
be approximated in practical calculations. A very successful approach which includes the exact-exchange and
dynamical correlation effects within the random phase
approximation is the GW method originally proposed
by Hedin.1 In this method, the self-energy is calculated
solely from the product of the Green function and the
dynamically screened Coulomb potential. Solving the
Dyson equation self-consistently using the self-energy
in this form gives the self-consistent GW quasi-particle
energies.2 In practice, under the assumption that the
non-interacting Kohn-Sham (KS) particles in densityfunctional theory (DFT) constitute a good zeroth order approximation to the quasi-particles, this self-energy
is often calculated using the Green function and the
screened Coulomb potential obtained from the KS eigenvalues and eigenfunctions.3,4 When the resultant selfenergy is treated as a correction to the xc potential of
the KS system, and the quasi-particle energies are given
by the corresponding corrections over the KS eigenvalues, the GW method simplifies into its most often used
form, i.e. the G0 W0 approximation.
In the last two decades, this scheme has achieved great
success in predicting/reproducing reasonable fundamental band gaps and band structures of semiconductors and
insulators.5–7 However, the results can depend on the KS
starting point, which reflects the limitations of this approximation. In recent years, several approximate selfconsistent GW methods have been proposed to overcome
the limitations of G0 W0 by mapping the non-hermitian
energy-dependent self-energy to a static hermitian effective potential. Applications to sp semiconductors and
a few d- and f -electron systems have shown promising
results.8–10 In any case, the quality of G0 W0 itself has
remained an open issue.
For simplicity and computational efficiency, the majority of calculations performed to date is based on
the pseudopotential method (the PP-G0 W0 method).5
Recent implementations within the full-potential allelectron framework (the AE-G0 W0 method), however,
have consistently revealed discrepancies between PPG0 W0 and AE-G0 W0 results.11–15 Understanding these
differences is of fundamental importance not only for a reliable description but also for an evaluation of the G0 W0
approximation itself. This is a prerequisite for a systematic development of methods to compute quasi-particle
energies beyond the G0 W0 approximation. Based on
these considerations, we analyze the discrepancies between the AE-G0 W0 and PP-G0 W0 results by addressing
the approximations underlying PP-G0 W0 . The fundamental band gaps in diamond, Si, BN, AlP, GaAs, LiF,
NaCl, and CaSe, and the semicore d-state binding ener-
2
gies in ZnS, ZnSe, ZnTe, CdS, CdSe, CdTe, and GaN are
taken as examples.
II.
METHODS
In the present study, we employ our recently developed computer code FHI-gap (Green function with augmented planewaves). This code represents an add on to
the full-potential (linearized-)augmented planewaves plus
local orbitals (FP-(L)APW+lo) package WIEN2k. The
method and details of FHI-gap are described in Ref. 16.
For a clear understanding of the following discussion, we
start below with the basic G0 W0 formalism. For a detailed description, we refer the reader to Ref. 6.
A.
The G0 W0 approximation
The calculation of the self-energy requires the knowledge of the non-interacting Green function, G0 , and the
screened Coulomb potential, W0 :
Z
i
0
Σ (r, r ; ω) =
G0 (r, r0 ; ω 0 + ω) W0 (r, r0 ; ω 0 ) dω 0 .
2π
(1)
G0 is represented in terms of the KS orbitals as
G0 (r, r0 ; ω) =
X [ϕnk (r0 )]∗ ϕnk (r)
n,k
ω − KS
nk − iη
,
(2)
where η = 0+ for occupied states (holes) and η = 0− for
unoccupied states (electrons). The dynamically screened
Coulomb potential is given by
Z
(3)
W0 (r, r0 ; ω) = ε−1 (r, r1 ; ω) v (r1 , r0 ) dr1 .
v (r, r0 ) is the bare Coulomb potential and ε (r, r0 ; ω) is
the dielectric function. It is calculated from
Z
ε (r, r0 ; ω) = 1 − v (r, r1 ) P0 (r1 , r0 ; ω) dr1 .
(4)
P0 (r, r0 ; ω) is the polarizability within the random-phaseapproximation (RPA), and it is evaluated using
Z
i
P0 (r, r0 ; ω) = −
G0 (r, r0 ; ω + ω 0 ) G0 (r0 , r; ω 0 ) dω 0 .
2π
(5)
To separate the exchange part of the self-energy, Σx ,
from the correlation, Σc = Σ − Σx , the bare Coulomb
potential is substracted from the screened one by W0c =
W0 − v. The exchange part of the self-energy is then
obtained by
Z
i
Σx (r, r0 ) =
G0 (r, r0 ; ω 0 ) v (r, r0 ) dω 0
2π
occ
(6)
X
ϕnk (r) v (r0 , r) ϕ∗nk (r0 ) ,
=
n,k
while the correlation part of the self-energy is given by
Z
i
c
0
Σ (r, r ; ω) =
G0 (r, r0 ; ω 0 + ω) W0c (r, r0 ; ω 0 ) dω 0 .
2π
(7)
The quasi-particle energy of the state, characterized by
the band index n and the reciprocal space vector k, is
obtained by taking the first-order correction over the KS
eigenvalue through
KS
0 qp
xc
qp
(r) |ϕnk i
nk = nk + hϕnk |< [Σ (r, r ; nk )] − V
(8)
with V xc being the xc potential of the KS particles.
B.
Levels of approximations underlying PP-G0 W0
To compute G0 W0 quasi-particle energies without further approximations requires an implementation of above
equations within a full-potential all-electron framework.
In this case, the correction to the KS eigenvalue in Eq. 8
is calculated through
qp
∆AE
nk = < (hϕnk |Σ ({ϕnk , ϕcore } , nk ) |ϕnk i) −
(9)
hϕnk |V xc (nval + ncore ) |ϕnk i .
Here, ϕ’s and n’s are the wavefunctions and densities of
the KS particles.
In the more often used PP-G0 W0 method, pseudopotentials are employed allowing for an efficient expansion
of P0 , ε, W0 , and Σ in a plane-wave basis. These pseudopotentials are generated from KS calculations of free
atoms. They are much softer than their full-potential
counterparts close to the nuclei and the divergence on
the nuclei is totally removed. In PP-DFT calculations
of polyatomic systems, Hartree and xc potential are
formally treated as functionals of the valence electron
density alone, with the pseudopotentials accounting for
the interactions of the valence electrons with core electrons and nuclei. This implies two major approximations. First, the core-valence xc potential is linearized
with respect to its dependence on ncore and nval . Second, the modifications of the core states due to the environment generated by the neighbouring atoms are discarded (frozen-core approximation). In addition, the
corresponding KS pseudopotential wavefunctions (density) are much smoother than their all-electron counterparts. In later discussions, we denote them as pseudowavefunctions (pseudo-density). The reliability of using
these pseudopotentials in DFT calculations depends on
the fulfilment of two conditions. First, the same xc functional should be used for generating the pseudopotential and performing the self-consistent calculation of the
polyatomic system.7,17 Second, the overlap of core- and
valence-electron densities shouldn’t be significant. When
this is not the case, like in alkali metals, non-linear core
corrections have to be added.18,19 A contrived core density needs to be carried along and included in the evaluation of the xc potential.
3
When the G0 W0 correction is applied, the self-energy
and the xc potential are calculated from the pseudowavefunctions, ϕ̃, and pseudo-density, ñ, for the valence
states only. Therefore, this correction reads
qp
∆PP
nk = <(< ϕ̃nk |Σ({ϕ̃nk }, nk )|ϕ̃nk >) −
< ϕ̃nk |V xc (ñval ) |ϕ̃nk > .
(10)
Compared with the AE-G0 W0 method (Eq. 9), there
are three approximations underlying PP-G0 W0 . The first
one is the frozen-core approximation. This means that
the core electrons stay in the states of the isolated atomic
calculation that were used when the pseudopotential was
constructed. In contrast, in all-electron calculations, the
core orbitals feel and adjust to the potential of the other
nuclei and the rearrangement of the valence electrons.
Second, since in the pseudopotential approximation the
self-energy is calculated only from the valence states,
the core-valence interaction is kept at the level of the
the xc potential with which the pseudopotential is generated. Analogous to the term “core-valence linearization” in DFT, we call this “core-valence partitioning”
in the G0 W0 calculation because the core-valence and
valence-valence interactions are treated independent of
each other and at different levels.15 Finally, there is also
an effect coming from the use of pseudo-wavefunctions,
which are much smoother than their full-potential counterparts (typically having removed all their nodes).
In this paper, we investigate the impact of these approximations by addressing them separately. Four sets
of calculations are performed to assess their influence
on the results. For the AE-G0 W0 calculations, we use
our newly developed G0 W0 code based on the fullpotential (linearized-)augmented planewave plus local orbitals method. Then, we fix the KS eigenvalues and
eigenfunctions of the core states at the atomic DFT level
and evaluate the G0 W0 corrections to the KS eigenvalues
(represented as AE-FC-G0 W0 in later discussions) by
qp FC
−
∆AE-FC
= < ϕnk |Σ ϕFC
nk
nk , ϕcore , nk |ϕnk
xc
FC
FC
ϕnk |V
nval + ncore |ϕnk .
(11)
Differences between results obtained from Eqs. 11 and 9
show the impact of the frozen-core approximation on the
G0 W0 corrections.
After this, we calculate the self-energy and xc potential
as functions of the all-electron valence wavefunctions and
density (denoted as AE-V-G0 W0 ) by
qp ∆AE-V
= < ϕnk |Σ ϕFC
−
nk
nk , nk |ϕnk
xc
FC
(12)
ϕnk |V
nval |ϕnk .
Comparing this equation with Eq. 11, we see the impact
of the core-valence partitioning. The results are, in the
end, compared with PP-G0 W0 calculations (Eq. 10) performed with the GWST code.20–22 That way, remaining
differences are traced back to the effect of using pseudowavefunctions.
III.
OPEN PROBLEMS
A.
The G0 W0 band gaps
In spite of the above mentioned deficiencies, for many
materials, PP-G0 W0 gives band gaps in good agreement
with experiments. The first few AE-G0 W0 calculations,
however, showed noticeably different results. For a fair
comparison of the two methods, we only consider results using the local-density approximation (LDA) for the
KS particles as the starting point (G0 W0 @LDA) unless
specifically mentioned. The PP-G0 W0 results agree better with experiments than their all-electron counterparts
in many cases.15 If we take Si as example, early reports
using the AE-G0 W0 method show band gaps of 0.85 eV
(Ref. 12) and 0.9 eV (Ref. 11), compared with the formerly reported PP-G0 W0 band gaps of 1.19 to 1.29eV.
Ku and Eguiluz attributed the better agreement of the
PP-G0 W0 results with experiments to an error cancellation between the absence of core electrons and lack of
self-consistency.12 Delaney and coworkers, in turn, ascribed the success of PP-G0 W0 to the error cancellation
between self-consistency and vertex corrections.23 They
further claimed that the discrepancy between the PPG0 W0 band gaps with the all-electron ones is due to the
lack of convergence with respect to the number of unoccupied bands in the all-electron calculations.24 Leaving
aside these arguments about the effects of self-consistency
and vertex corrections, Tiago et al. performed a set of
all-electron like PP-G0 W0 calculations for Si, GaAs, and
Ge including the outermost two shells (one valence shell
and one ”core” shell) as valence states.13 They reported
a value of 1.04 eV for the band-gap of Si when convergence with respect to the number of unoccupied states is
achieved. This number is supported by a well-converged
value of 1.00 eV obtained by the projector-augmented
wave (PAW) method (Ref. 25), and FP-(L)APW+lo results of 1.05 eV (Ref. 14) and 1.00 eV (Ref. 15). This
analysis suggests that the band gap reported in Ref. 12 is
only converged roughly 0.1 eV with respect to the number of unoccupied states. However, the reason for the
mismatch between the well-converged AE-G0 W0 band
gaps and the previous PP-G0 W0 ones is still an interesting and important question to be investigated. In our
previous letter (Ref. 15), this issue has been addressed
for several distinct crystals (diamond, Si, BN, AlP, LiF,
NaCl, CaSe, and GaAs). It was revealed that the corevalence partitioning effects on the exchange part of the
self-energy and the xc potential are strong, but tend to
cancel each other. For materials without semicore states
(diamond, Si, BN, AlP, and LiF), discrepancies in the order of 0.1 eV remain. This justifies the PP-G0 W0 @LDA
method in obtaining the band gaps of such systems with
an accuracy of the order of 0.1 eV. For materials with
semicore states (NaCl, CaSe, and GaAs), this inaccuracy
turned out to be larger. In addition to these findings, two
questions remain to be addressed. What is the effect of
the frozen-core approximation? To which extent can the
4
core-valence partitioning effects be reduced when including further shells as valence states?
B.
Semicore d-state binding energies
IIB -VI semiconductors and group-III nitrides are technologically important materials in optical applications
due to their wide band gaps.26 The semicore d states
from the cation play a crucial role in the chemical bonding. Already in the 1980’s, all-electron calculations had
shown that neglecting effects of the semicore d states on
the chemical bonding leads to a wrong description of the
ground state properties.27–29
In spite of this fact, Zakharov et al. performed the
first quasi-particle band-structure calculations for this
kind of materials employing the pseudopotential method
and treating the d states as core states.30 Later, Rohlfing, Krüger, and Pollmann31 showed that the reasonable
agreement with experiments obtained by Ref. 30 was, in
fact, spurious. Furthermore, they pointed out that including the d states in the valence configuration worsens
the results, and an explicit treatment of the whole shell of
semicore states (e.g. the third shell of Ga in GaN) is necessary for getting reliable band gaps and d band positions
and widths. This trend has been confirmed by further
calculations using PP-G0 W0 .32–34 Recently, Rinke et al.
reported excellent agreement with experiments using PPG0 W0 including only the d states in the valence region.7
Different from the standard treatment, these calculations
were based on the optimized effective-potential approach
together with LDA correlation (denoted as OEPx +cLDA
in later discussion). These findings highlight the fact that
different starting points on the KS level and orbitals and
different core-valence interactions can induce very different results compared to those of the above mentioned
PP-G0 W0 @LDA studies.
Although AE-G0 W0 calculations have been carried out
for a number of materials including some of the IIB -VI
semiconductors and group III nitrides,11,35 a systematic
study of the electronic structure of these materials and
the role of core-valence interactions within the AE-G0 W0
framework is still lacking. We also note that discrepancies up to 0.5 eV for the d-state binding energies still remain between the PP-G0 W0 and AE-G0 W0 results. As
semicore d states couple to the core states differently from
the valence band maximum (VBM) and conduction band
minimum (CBM), core-valence partitioning effects are to
be expected.
IV.
A.
RESULTS
Band gaps in sp semiconductors
In Ref. 15, we have analyzed the discrepancy between
the AE-G0 W0 and PP-G0 W0 band gaps by separating
the errors from the core-valence partitioning and the use
FIG. 1: Errors in the G0 W0 band gap corrections of Si, diamond, BN, AlP, and LiF arising from the core-valence partitioning (orange), the use of pseudo-wavefunctions (purple),
and the frozen-core approximation (green). They are computed from the differences between Eqs. 12 and 11, Eqs. 10
and 12, as well as Eqs. 11 and 9, respectively. The black bars
show the sum of these contributions.
of pseudo-wavefunctions in Si, diamond, BN, AlP, LiF,
NaCl, CaSe, and GaAs. Special emphasis was put on
three examples: Si, NaCl, and GaAs. In this Section,
we provide a more systematic discussion of this issue by
categorizing the materials in terms of two groups. Materials without semicore states will form the first set of five
examples. Errors from the core-valence partitioning and
the use of pseudo-wavefunctions will be addressed. In
addition, the frozen-core approximation, a factor which
hasn’t been discussed yet, will be studied. The corresponding PP-G0 W0 and AE-V-G0 W0 calculations treat
only the outermost shell as valence. Since in the AE-VG0 W0 calculations the valence electrons feel the potential with the core states frozen (Eq. 12), small differences
in the wavefunctions from those of the general AE-G0 W0
approach used in Ref. 15 (where core states were allowed
to relax) result in changes of the order of 0.01 eV. Materials with semicore states will form the second set of three
examples. It was shown in Ref. 15 that the existence
of semicore states (Na 2p in NaCl, Ga 3d and As 3d in
GaAs, and Ca 3p in CaSe) makes the core-valence partitioning inappropriate. Accordingly, we include the shell
of these semicore states into the valence configuration.
This way, we can check if the conclusions we draw for
materials without semicore states persist for those with
semicore states when these semicore shells are treated as
valence.
The experimental lattice constants of 5.43 Å (Si), 3.57
Å (diamond), 3.62 Å (BN), 4.02 Å (LiF), 5.45 Å (AlP),
5.63 Å (NaCl), 5.91 Å (CaSe), and 5.66 Å (GaAs) are
used throughout this paper. Using experimental geometries is appropriate and, in fact, necessary because we like
to compare our theoretical results to experimental band
gaps without unwanted problems caused by differences in
5
the lattice constants. The integration over the Brillouin
zone (BZ) was carried out with a 4×4×4 mesh, ensuring
convergence within 0.01 eV for the band-gap correction
with respect to the number of sampling points, except in
diamond and GaAs, where a 6 × 6 × 6 mesh was used
instead. Convergence within 5 meV with respect to the
number of included excited states is achieved for all the
materials. For example, in Si (NaCl) we found that about
150 (300) unoccupied bands are required.
In Fig. 1, we show the contributions from the frozencore approximation, the core-valence partitioning, and
the use of pseudo-wavefunctions to the differences between AE-G0 W0 and PP-G0 W0 band-gap corrections in
Si, diamond, BN, AlP, and LiF. It clearly reveals that
errors from the core-valence partitioning and the use of
pseudo-wavefunctions account for most of the discrepancies between the AE-G0 W0 and PP-G0 W0 band gaps.
For a detailed analysis, we show in Tables I, II, and III
the matrix elements of the self-energy and the xc potential for the highest occupied and the lowest unoccupied states as well as their differences from the AEG0 W0 , AE-FC-G0 W0 , AE-V-G0 W0 , and PP-G0 W0 calculations in diamond, AlP, and LiF, respectively. Taking
AlP as example, the CBM is at the X point and the
VBM at Γ, labeled Xc and Γv , respectively. The difference of V xc between these two points is denoted as
∆V xc =< ϕXc |V xc |ϕXc > − < ϕΓv |V xc |ϕΓv >. ∆Σx
and ∆Σc are defined analogously. We find the following
general trends:
• The error introduced by the frozen-core approximation increases with decreasing energy distance between the highest core state and the VBM. In other
words, the core states need to be tightly bound to
justify the frozen-core approximation.
• The effect of core-valence partitioning on the correlation part of the self-energy and its contribution
to the band-gap correction is in the order of 0.01
eV and, therefore, negligible.
• The largest error appears in the exchange part of
the self-energy, Σx , and the exchange part of the
xc potential, V x . These errors are due to the corevalence partitioning, and they are of the order of 1
eV.
• These core-valence partitioning errors tend to cancel each other in their contribution to the bandgap correction. This is reflected by the difference
between ∆Σx − ∆V xc in AE-FC-G0 W0 and AE-VG0 W0 calculations. This value is 0.26 eV in diamond, −0.01 eV in AlP, and 0.24 eV in LiF.
• Summing up these errors from ∆Σx − ∆V xc with
those from ∆Σc , the core-valence partitioning increases the band-gap corrections. However, this increase is not enough to solely explain the discrepancy between AE-G0 W0 and PP-G0 W0 results.
• The use of pseudo-wavefunctions plays an equally
important role in the difference between AE-G0 W0
and PP-G0 W0 band gaps.
Let us have a closer look at the error from the frozencore approximation as this has not been addressed in
Ref. 15. Figure 1 indicates that it is below numerical
accuracy in diamond, Si, and BN. In AlP and LiF, it is
−0.01 eV. We observe that it increases with decreasing
energy distance between the highest core state and the
VBM. For example, in BN, the KS eigenvalues of the
core states are 170 eV below the VBM. These states are
tightly bound to the nuclei. The errors from the frozencore approximation are negligible in both the LDA and
G0 W0 calculations. The error in the matrix elements of
Σc , Σx , and V xc is smaller than 0.01 eV. In AlP, the
KS eigenvalues of the Al 2p states are 63.9 eV below the
VBM. In FP-(L)APW+lo based DFT calculations, these
states are normally treated as valence states. When they
are considered as core states and frozen, the error in the
LDA band gap is still negligible, but that in the G0 W0
correction is not. The maximum change in the matrix
elements of Σc , Σx , and V xc is 0.04 eV. It appears in Σx
of the lowest unoccupied state at the X point (-5.31 eV
compared with −5.27 eV in the forth column of Table II).
This error in the self-energy is a clear indication that the
self-energy is more sensitive to the slight variation of the
wavefunctions due to the frozen-core approximation than
the xc potential. In LiF, the KS eigenvalue of the Li 1s
state is 39.6 eV below the VBM. Here, the error from
the frozen-core approximation is already observable in
the KS band gap, which changes by 0.02 eV. The maximum error in the matrix elements is 0.09 eV. It shows up
in Σx of the lowest unoccupied state Γc (-7.71 eV compared with -7.62 eV in the forth column of Table III).
This error, is fortunately canceled by the corresponding
change in the correlation part of the self-energy, resulting
in a final impact of only −0.01 eV on the G0 W0 band-gap
correction. In summary, the frozen-core approximation
requires for being valid that the core states are tightly
bound. In Si, diamond, and BN, where this is the case,
the error from this approximation is negligible. In AlP
and LiF, these errors become non-negligible. The selfenergies are more sensitive to this approximation than
the xc potential.
Let us turn now to the second set of materials, i.e.,
those with semicore states. In Ref. 15, the matrix elements of Σc , Σx , and V xc in NaCl and GaAs were analyzed for the case when only the outermost shell was
treated as valence. Due to the presence of semicore
states, incomplete error cancellation between ∆Σx and
∆V xc was observed which led to large and negative corevalence partitioning errors. The higher in energy the
semicore states are, the larger core-valence partitioning
errors become. Going from NaCl through CaSe to GaAs,
where the center of the semicore bands (at the KS level)
is 20.8 eV, 18.6 eV, and 14.8 eV below the VBM respectively, the corresponding core-valence partitioning errors
are −0.09 eV, −0.22 eV, and −0.45 eV.15 Since the semi-
6
TABLE I: The matrix elements of the self-energy (correlation part Σc , exchange part Σx ) and of the xc potential V xc (correlation
part V c , exchange part V x ) for diamond, obtained from AE-G0 W0 , AE-FC-G0 W0 , AE-V-G0 W0 , and PP-G0 W0 calculations.
All quantities are given in eV and are shown for the valence band maximum, Γv , and the conduction band minimum, CBM. ∆
indicates that the difference between the quantities at these two points is taken.
AE-G 0 W 0
Γv
CBM∗
∆
AE-FC-G 0 W 0
Γv
CBM∗
∆
AE-V-G 0 W 0
Γv
CBM∗
∆
PP-G 0 W 0
Γv
CBM∗
∆
∗
LDA
Σc
Σx
Vc
Vx
V xc
Σx − V xc
Σ − V xc
G0 W0
4.10
1.45
-4.10
-6.55
-20.12
-9.77
10.35
-1.80
-1.67
0.13
-16.20
-13.85
2.35
-18.00
-15.52
2.48
-2.12
5.75
7.87
-0.67
0.65
1.32
5.42
4.10
1.45
-5.10
-6.55
-20.12
-9.77
10.35
-1.80
-1.67
0.13
-16.20
-13.85
2.35
-18.00
-15.52
2.48
-2.12
5.75
7.87
-0.67
0.65
1.32
5.42
5.10
1.49
-5.12
-6.61
-19.15
-8.70
10.45
-1.77
-1.65
0.12
-15.00
-12.80
2.20
-16.77
-14.45
2.32
-2.38
5.75
8.13
-0.89
0.63
1.52
5.62
4.15
1.24
-5.20
-6.44
-19.15
-8.67
10.48
-16.81
-14.30
2.51
-2.34
5.63
7.97
-1.10
0.43
1.53
5.68
In diamond the conduction band minimum, CBM, lies at 66.7 % of the distance between Γ and X, which is a sampling
point of the 6 × 6 × 6 mesh.
TABLE II: Same as Table I for AlP with the CBM at the X point.
AE-G 0 W 0
Γv
Xc
∆
AE-FC-G 0 W 0
Γv
Xc
∆
AE-V-G 0 W 0
Γv
Xc
∆
PP-G 0 W 0
Γv
Xc
∆
LDA
Σc
Σx
Vc
Vx
V xc
Σx − V xc
Σ − V xc
G0 W0
1.44
1.42
-3.96
-5.38
-16.08
-5.27
10.81
-1.58
-1.31
0.27
-12.49
-8.09
4.40
-14.07
-9.40
4.67
-2.01
4.13
6.14
-0.59
0.17
0.76
2.20
1.44
1.42
-3.95
-5.37
-16.10
-5.31
10.79
-1.58
-1.31
0.27
-12.49
-8.09
4.40
-14.07
-9.40
4.67
-2.03
4.09
6.12
-0.61
0.14
0.75
2.19
1.44
1.40
-3.94
-5.34
-14.08
-4.48
9.60
-1.53
-1.29
0.24
-10.50
-7.25
3.25
-12.03
-8.54
3.49
-2.05
4.06
6.11
-0.65
0.12
0.77
2.21
1.47
1.34
-3.82
-5.16
-14.09
-4.57
9.52
-11.82
-8.49
3.33
-2.27
3.92
6.19
-0.93
0.10
2.50
1.03
core states of these materials are high in energy, it is
impossible to disentangle the effects of the frozen-core
approximation and the core-valence partitioning.
In Ref. 13, it was illustrated that the PP-G0 W0 results
for GaAs can be improved by including the whole shell of
the semicore states into the valence region. We therefore
now investigate the effects of the frozen-core approximation and the core-valence partitioning when the semicore
shells are treated as valence. In NaCl and CaSe, this
means treating the outermost two shells of the cations as
valence shells. In GaAs, the same treatment applies to
both Ga and As atoms. In Fig. 2, we shown the errors
from the frozen-core approximation and the core-valence
partitioning in these three materials when the semicore
shells are treated as valence. All the conclusions we have
drawn above about the core-valence partitioning for materials without semicore states turn out to be valid also
here. The impact of the frozen-core approximation is
negligible in NaCl and GaAs, where the KS eigenvalues
of the highest core states are more than 181 eV below the
VBM. In CaSe, the KS eigenvalues of the Se 3d states are
only 46.0 eV below the VBM. Similar to what we have
observed in LiF, errors from the frozen-core approximation become non-negligible. The largest discrepancy is
due to the exchange part of the self-energy (-17.41 eV
compared with −17.60 eV in the forth column of Table IV). In this case, this error is not fully canceled by
the corresponding change in the correlation part of the
7
TABLE III: Same as Table I for LiF with the CBM at the Γ point.
AE-G 0 W 0
Γv
Γc
∆
AE-FC-G 0 W 0
Γv
Γc
∆
AE-V-G 0 W 0
Γv
Γc
∆
PP-G 0 W 0
Γv
Γc
∆
LDA
Σc
Σx
Vc
Vx
V xc
Σx − V xc
Σ − V xc
G0 W0
8.97
5.50
-3.42
-8.92
-31.87
-7.62
24.25
-1.97
-1.45
0.52
-22.26
-11.70
10.56
-24.23
-13.15
11.08
-7.64
5.53
13.17
-2.14
2.11
4.25
13.22
8.99
5.47
-3.37
-8.84
-31.87
-7.71
24.16
-1.97
-1.45
0.52
-22.26
-11.70
10.56
-24.23
-13.15
11.08
-7.64
5.44
13.08
-2.17
2.07
4.24
13.23
8.99
5.49
-3.37
-8.86
-30.65
-5.47
25.18
-1.94
-1.39
0.55
-21.03
-9.74
11.29
-22.97
-11.11
11.86
-7.68
5.64
13.32
-2.19
2.27
4.46
13.45
8.79
4.45
-3.68
-8.13
-30.43
-5.77
24.66
-22.81
-11.05
11.76
-7.62
5.28
12.90
-3.17
1.60
4.77
13.56
TABLE IV: Same as Table II for CaSe, but without PP-G0 W0 results.
AE-G 0 W 0
Γv
Xc
∆
AE-FC-G 0 W 0
Γv
Xc
∆
AE-V-G 0 W 0 (Ca 2s, 2p, 3s; Se 4s, 4p)
Γv
Xc
∆
LDA
Σc
Σx
Vc
Vx
V xc
Σx − V xc
Σ − V xc
G0 W0
1.86
1.87
-4.06
-5.93
-17.60
-5.70
11.90
-1.61
-1.39
0.22
-13.64
-9.40
4.24
-15.25
-10.79
4.46
-2.35
5.09
7.44
-0.48
1.03
1.51
3.37
1.86
1.81
-4.04
-5.85
-17.41
-5.71
11.70
-1.61
-1.39
0.22
-13.63
-9.40
4.23
-15.24
-10.79
4.45
-2.17
5.08
7.25
-0.36
1.04
1.40
3.26
1.86
1.82
-4.07
-5.89
-14.69
-5.12
9.57
-1.54
-1.38
0.16
-10.76
-8.81
1.95
-12.30
-10.19
2.11
-2.39
5.07
7.46
-0.57
1.00
1.57
3.43
self-energy. The overall impact of the frozen-core approximation is a reduction of the band gap. Summing
up with the associated affect from the core-valence partitioning which, in contrast, increases the gap, we can
conlcude that these approximations (mimicked by AE-VG0 W0 ) typically lead to an overestimation of band gaps
when semicore shells are included in the valence configuration. As a matter of fact, all conclusions we have drawn
above about core-valence partitioning in materials without semicore states become valid again.
B.
Band gaps and semicore d states in IIB -VI
semiconductors and group-III nitrides
In this Section, we compare our AE-G0 W0 results for
IIB -VI semiconductors and group-III nitrides with experiments and other theoretical results. To investigate
the core-valence interactions on different levels, the corevalence partitioning effects are analyzed using different
configurations of valence states. We restrict ourselves
to the metastable zinc-blende structure and experimen-
tal lattice constants. The BZ integrations were carried
out with a 6 × 6 × 6 mesh in all cases, and ∼150 unoccupied states were included to ensure the convergence
of the fundamental band gaps (d-state binding energies)
within 0.01 eV (0.03 eV) with respect to the number of
unoccupied states.
In Table V, we show the band gaps together with other
theoretical and experimental data. Since spin-orbit interaction is not considered in our calculations, the VBM
here corresponds to the Γ15v state of the zinc-blende crystal. Including spin-orbit interaction would split it up by
the spin-orbit coupling constant ∆0 into a Γ8v state and a
Γ7v state with the corresponding energy values increased
by ∆0 /3 and lowered by 2∆0 /3, respectively. Therefore,
the band gap obtained for the Γ15v state is overestimated
by ∆0 /3. For comparison with experiment, this value
should, hence, be corrected.11,36 Such comparison confirms the general trend we have observed in the previous
section. The PP-G0 W0 method (with the outermost two
shells treated as valence) gives larger band gaps than AEG0 W0 . In fact, the AE-G0 W0 band gaps remain seriously
smaller than the experimental values. On the other hand,
8
FIG. 2: Errors in the band gaps of NaCl, CaSe, and GaAs
arising from the core-valence partitioning (orange) and the
frozen-core approximation (green) when the semicore shells
are treated as valence. The black bars show the respective
sums of these contributions.
our results are in excellent agreement with the AE-G0 W0
values of van Schilfgaarde, Kotani, and Faleev,11,35 using
the linear muffin-tin orbital (LMTO) method. Differences larger than 0.15 eV are only observed in GaN and
CdS, which we assign to the fact that they have used the
wurtzite structure which is known to exhibit larger band
gaps than the zinc-blende phase.37,38
Table VI shows our results for the semicore d-state
binding energies. The presented values refer to the average position of the semicore d states at Γ with respect to
the VBM.36 Taking GaN as an example, there are three
and two degenerate d states, respectively, at −13.58 eV
and −13.27 eV below the VBM at the LDA level. Hence,
the LDA d-state binding energy of 13.46 eV given in Table VI is obtained by taking the average over these values
considering the corresponding weights of 3 and 2. As, like
mentioned above, spin-orbit coupling is not considered
here (the Γ15 state is taken as reference rather than the
Γ8v state), one should add ∆0 /3 to the computed value
for comparison with experiment. (Likewise, in Refs. 36
and 11 ∆0 /3 was subtracted from the experimental data.)
In Ref. 36, it was pointed out that the use of the
plasmon-pole approximation for the screening as done
in Ref. 32 may introduce additional errors in the band
structure. Thus, we consider the values reported in Ref.
36 as the state-of-the-art PP-G0 W0 @LDA results. Indeed, these values agree within 0.1 eV with our AEG0 W0 data. Both of them differ considerably (∼0.5 eV)
from the all-electron values of Ref. 11. Therefore, the
formerly reported discrepancies between the LDA based
AE-G0 W0 and PP-G0 W0 results are reduced from up to
0.5 eV to less than 0.1 eV. Overall, the G0 W0 method represents a substantial and systematic improvement over
LDA. However, discrepancies with experiments, scattering from ∼ 1.3 eV (CdSe) to ∼ 2.3 eV (ZnTe), still remain.
To analyze the role of core-valence interaction, we compare the band structures of GaN and ZnS obtained by
AE-V-G0 W0 (Eq. 12) using different configurations of
valence states.
Since the 3s and 3p states of the IIB -VI semiconductors and group-III nitrides belong to the core states, we
call the n = 1 and n = 2 shells of these materials the
deep core. In GaN, these deep core states include Ga 1s,
2s, and 2p and N 1s. In ZnS, the deep core states comprise the n = 1 and n = 2 shells of both Zn and S. The
first set of AE-V-G0 W0 calculations treats the deep core
states plus the 3s and 3p states of Ga and Zn as core.
The 3d, 4s, and 4p states of the cation and the outermost
shell of the anion are considered as valence. The second
set includes only the deep core states in the core. The
n = 3 shell of both Ga and Zn plus the outermost shell
of the anion are treated as valence. These two configurations are denoted as AE-V1-G0 W0 , and AE-V2-G0 W0 ,
respectively.
The comparison between AE-V1-G0 W0 and AE-V2G0 W0 results allows the study of the interaction between
the electrons within the semicore shell of the cation. The
further difference between AE-V2-G0 W0 and AE-G0 W0
results reveals the influence of the deep core states.
In Figs. 3 and 4, we display the band diagrams of GaN
and ZnS from the LDA, AE-V1-G0 W0 , AE-V2-G0 W0 ,
and AE-G0 W0 calculations. Including only the outermost shell and the d states of the cation in the G0 W0
correction (AE-V1-G0 W0 ) moves the d states up in energy by several eV’s. In GaN, the Ga 3d states jump out
of the energy range of the N 2s band. This is found between −13 and −17 eV in the left panel of Fig. 3, while
the bands at ∼ −10 eV correspond to the Ga 3d states.
In ZnS, they even enter the region of the sulfur-derived
valence p bands, as can be seen from Fig. 4. When the interaction between the 3d states and the 3s and 3p states
(the semicore shell) in the cation is included in the G0 W0
calculation (AE-V2-G0 W0 ), these bands shift back in the
right direction toward the AE-G0 W0 results. This shows
that the inconsistent treatment of the interactions between the cation semicore d states, sp states in the same
shell, and the anion sp valence states is responsible for
the unphysical lowering of these d-state binding energies.
Nevertheless, discrepancies in the order of 1 eV with respect to the AE-G0 W0 results still remain, a clear indication that also the interaction with the deep core states
is necessary for a correct description of the semicore d
states. In our calculations, we observe that removing the
deep core states for the calculation of the polarizability
produces a negligible effect. This is in agreement with
the expectation that the deep core states do not participate in the screening. However, their interaction with
other states, especially the semicore d states, through
the screened Coulomb potential is still important.
To analyze the origin of these differences, we follow
a procedure analogous to that employed in the previous
section for the band gaps. In Table VII, we show the matrix elements of the self-energy and the xc potential at
9
TABLE V: G0 W0 band gaps for GaN, ZnS, ZnSe, ZnTe, CdS, CdSe, and CdTe in the zinc-blende structure. Experimental and
theoretical results from the literature are shown for comparison. All quantities are in eV. LDA values are given in parentheses.
Lattice constant in Å
∆0 /3 [39]
This work
G0 W0 @LDA
AE-G 0 W 0 results from
G0 W0 @LDA [11, 35]
PP-G 0 W 0 results from
G0 W0 @LDA [32]
G0 W0 @LDA [36]
PP-G 0 W 0 results from
G0 W0 @OEPx +cLDA [7]
Experiment [39, 40]
GaN
4.50
0.00
ZnS
5.40
0.02
ZnSe
5.67
0.13
ZnTe
6.09
0.32
CdS
5.82
0.02
2.79 (1.76)
3.19 (1.85)
2.36 (1.00)
2.26 (1.02)
1.85 (0.87)
literature
3.03 (1.81)
3.21 (1.86)
2.25 (1.05)
2.23 (1.03)
1.98 (0.93)
literature (outermost two shells as valence)
2.88
3.50
2.45
3.41 (1.84)
2.37 (1.02)
2.27 (1.04)
2.13 (0.82)
literature (semicore d states plus outermost shell as valence)
3.09
3.62
2.36
3.30
3.87
2.95
2.68
2.50
CdSe
6.05
0.14
CdTe
6.48
0.32
1.22 (0.35)
1.36 (0.50)
1.37 (0.51)
1.38 (0.29)
1.51 (0.49)
1.83
1.90
TABLE VI: G0 W0 d-state binding energies (in eV) for zinc-blende GaN, ZnS, ZnSe, ZnTe, CdS, CdSe, and CdTe. LDA values
are displayed in parentheses.
GaN
ZnS
ZnSe
This work
G0 W0 @LDA
15.76 (13.46) 6.84 (6.35) 7.12 (6.58)
AE-G 0 W 0 results from literature
G0 W0 @LDA [11])
16.40 (13.60) 7.10 (6.20) 7.70 (6.70)
PP-G 0 W 0 results from literature (outermost two shells as valence)
G0 W0 @LDA [32]
15.70
6.40
G0 W0 @LDA [36]
6.84 (6.31) 7.17 (6.55)
PP-G 0 W 0 results from literature (semicore d states plus outermost
G0 W0 @OEPx +cLDA [7]
16.12 (14.85) 6.97 (6.91)
Experiment [39, 41, 42, 43] 17.1
8.7
9.2
the VBM and the highest d state at the Γ point (Γd ) for
GaN, as well as the difference between them. From the
sign of ∆Σc and ∆Σx , we see that the exchange interaction between the semicore d states, the s and p states in
the same shell, and the deep core states tends to increase
the binding energy of the d states, while the corresponding correlation interaction tends to reduce it. The correlation part of the self-energy on the VBM, mainly a 2p
state of N, shows very small differences between AE-V1G0 W0 , AE-V2-G0 W0 , and AE-G0 W0 calculations (< 0.1
eV), indicating that the correlation interaction between
this state and the 3s and 3p states of the cation as well
as the deep core states is weak. However, looking at
this term for the highest d state, we observe that the
correlation interaction between the 3s, 3p, and 3d states
is very strong, moving the semicore d states up by ∼2
eV when going from the AE-V1-G0 W0 to AE-V2-G0 W0
calculations. This is due to the strong overlap between
wavefunctions of electrons in the same shell. Explicitly
including these interactions at the G0 W0 level, the semicore d band moves down in energy due to a stronger
contribution from ∆Σx . Thus, treating all interactions
in the semicore shell at the G0 W0 level is obviously necessary. Including the deep core states further increases
the correlation part of the self-energy by 0.30 eV (∆Σc of
ZnTe
CdS
CdSe
CdTe
7.55 (6.99)
8.16 (7.63)
8.45 (7.86)
8.74 (8.18)
8.40 (7.72)
8.79 (8.08)
9.7
10.5
8.20 (7.50)
8.10
7.60 (6.69) 8.13 (7.53)
shell as valence)
7.66 (7.57)
9.84
9.5
TABLE VII: Matrix elements of the self-energy in eV (correlation part: Σc , exchange part: Σx ) and xc potential V xc
(correlation part: V c , exchange part: Σx ) for GaN in our
AE-V1-G0 W0 , AE-V2-G0 W0 , and AE-G0 W0 calculations.
LDA
Σc
AE-V1-G 0 W 0 (Ga
Γv
3.16
Γd
8.51
∆
13.26
-5.35
AE-V2-G 0 W 0 (Ga
Γv
3.22
Γd
10.23
∆
13.26
-7.01
AE-G 0 W 0
Γv
3.19
Γd
10.50
∆
13.26
-7.31
Σx
V xc
Σ − V xc
3d, 4s, 4p)
-21.60
-18.65
0.21
-38.28
-33.28
3.51
16.68
14.63
-3.30
3s, 3p, 3d, 4s, 4p)
-22.62
-19.21
-0.19
-51.90
-40.11
-1.56
29.28
20.90
1.37
-23.88
-56.81
32.93
-20.48
-43.76
23.28
-0.21
-2.55
2.34
G0 W0
9.96
14.63
15.60
−7.31 in AE-G0 W0 compared to −7.01 in AE-V2-G0 W0
in Table VII). This discrepancy is much smaller than in
the previous case but still not negligible.
The matrix elements for the exchange part of the selfenergy (forth column in Table VII) for the VBM increase
by almost 1 eV by including the 3s and 3p states (going
10
TABLE VIII: Same as Table VII for ZnS.
LDA
Σc
AE-V1-G 0 W 0 (Zn
Γv
2.41
Γd
8.00
∆
6.06
-5.59
AE-V2-G 0 W 0 (Zn
Γv
2.55
Γd
9.85
∆
6.06
-7.30
AE-G 0 W 0
Γv
2.58
Γd
10.13
∆
6.06
-7.55
FIG. 3: Band structure of cubic GaN from G0 W0 (black circles) compared to LDA (solid gray lines). The top of the
valence band is chosen as zero energy. The panels from left
to right correspond to the AE-V1-G0 W0 , AE-V2-G0 W0 , and
AE-G0 W0 calculations, respectively.
FIG. 4: Same as Fig. 3 for ZnS.
from AE-V1-G0 W0 to AE-V2-G0 W0 ) and another 1.3 eV
by including the deep core states (going from AE-V2G0 W0 to AE-G0 W0 ). On the other hand, the same matrix elements for the d state show huge changes (∼13
eV when including the 3s and 3p states of Ga and further 5 eV when including the deep core electrons of Ga
and N). These changes are carried along to the binding
energy of the 3d states. Summing up all these contributions, we obtain ∼11 eV from the 3s and 3p states of Ga
and another ∼3 eV coming from the deep core states in
the exchange part of the self-energy. The corresponding
contributions from the xc potential are 6.27 eV (change
of ∆V xc going from AE-V1-G0 W0 to AE-V2-G0 W0 in
Table VII) and 2.38 eV (change of ∆V xc going from AEV2-G0 W0 to AE-G0 W0 in Table VII), respectively. The
Σx
V xc
3d, 4s)
-18.44
-16.36
-34.21
-30.04
15.77
13.68
3s, 3p, 3d, 4s)
-20.34
-17.38
-46.61
-36.40
26.27
19.02
-23.08
-50.87
27.79
-19.93
-39.65
19.72
Σ − V xc
G0 W0
0.33
3.83
-3.50
2.56
-0.41
-0.36
-0.05
6.01
-0.57
-1.09
0.52
6.58
correction to the binding energy is again much smaller,
resulting from an error cancellation between the contributions from the self-energy and the xc potential. All
in all, the change in the G0 W0 correction to the d band
position going from AE-V1-G0 W0 to AE-V2-G0 W0 , and
from the AE-V2-G0 W0 to AE-G0 W0 is as much as 4.67
eV and 0.97 eV, respectively (last column in Table VII).
¿From this analysis we conclude that the exchange interaction of the 3s and 3p with the 3d states of Ga is
the main reason for the necessity of including all these
states in the G0 W0 calculations in order to obtain reliable results for the binding energy of the d states. This is
in agreement with what was found by Rohlfing, Krüger,
and Pollmann in Ref. 31. However, the correlation interaction between these states produces large corrections of
∼2 eV which can also not be neglected. The deep core
states contribute to the d-state binding energies through
both exchange and correlation. The total contribution
increases the d-state binding energies. Unlike what was
found in the previous section for the band gaps, where
the errors from core-valence partitioning of ∆Σ and ∆V xc
cancel, the error cancellation between these two quantities for the d state position is incomplete (difference of
0.97 eV, when comparing ∆(Σ − V xc ) from the AE-V2G0 W0 and AE-G0 W0 calculations in Table VII). On the
other hand, the corresponding PP-G0 W0 results in Table VI are similar to our AE-G0 W0 data. We therefore
conclude that a large error stemming from the use of
pseudo-wavefunctions must exist, which fortunately cancels the one arising from core-valence partitioning.
In Table VIII, we show the same analysis as before, but
for ZnS. Similar conclusions as for GaN can be drawn.
The exchange interaction between the core states (the
deep core states plus the 3s and 3p states of the cation)
and the semicore d states increases the d-state binding energy, while the correlation interaction decreases it. The
dominant core-valence partitioning effects appear in the
exchange part of the self-energy. They are partly canceled by the large changes in the exchange-correlation
potential and the small but non-negligible changes in the
correlation part of the self-energy. Upon going from AEV1-G0 W0 to AE-V2-G0 W0 , and from AE-V2-G0 W0 to
11
AE-G0 W0 (last column in Table VIII), we find that the
total G0 W0 correction to the d-state binding energy increases by 3.45 eV and 0.57 eV, respectively. The latter
shows that also the deep core states contribute significantly to the semicore d-state binding energy.
Let us finally briefly address the issue of the starting point underlying the G0 W0 calculation. In Ref. 7,
OEPx +cLDA has been used in both generating the pseudopotentials as well as the self-consistent DFT calculations. Only the outermost shell and the semicore d states
were treated as valence. Therefore, according to our analysis above, we expect the correlation interactions, which
tend to decrease the semicore d-state binding energies, to
be seriously underestimated. The corresponding results
for band gaps and semicore d-state binding energies are
shown in Tables V and VI, respectively. We observe two
trends: (i) The so calculated band gaps overall agree better with experiment than the AE results. (ii) The semicore d band positions are comparable to our AE values.
We conclude that the G0 W0 @OEPx +cLDA calculations
benefit from the exclusion of the core-valence correlation
interaction. Equally important is the starting point because perturbation theory is better justified when the KS
potential V xc is closer to the self-energy Σ.
V.
CONCLUSIONS
In this paper, we have analyzed the differences between all-electron G0 W0 and pseudopotential G0 W0 by
studying a series of sp semiconductors, IIB -VI semiconductors, and group-III nitrides. To explain discrepancies
between AE-G0 W0 and PP-G0 W0 band gaps, approximations underlying PP-G0 W0 , i.e., the frozen-core approximation, the core-valence partitioning, and the use
of pseudo-wavefunctions were separately addressed. The
frozen-core approximation is only justified when the core
states are tightly bound. In such cases, it has a negligible influence (<0.01 eV) on the band gaps. Otherwise, the self-energy is sensitive to slight changes in the
wavefunctions, and the final impact of the frozen-core approximation on the band gaps becomes visible. Effects
∗
1
2
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Permanent address: Peking University, 100871, Beijing,
China
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PP-G0 W0 calculations obviously benefit from a fortunate
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can however, not be considered as guaranteed. A reliable description of many-body interactions at the G0 W0
level, hence, requires the performance of all-electron calculations.
Acknowledgments
We thank Patrick Rinke, Xinguo Ren, Christoph
Freysoldt, and Martin Fuchs for helpful discussions. Financial support from the Nanoquanta NoE and the Austrian Science Fund is appreciated.
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